Example 29.1 Linear Response Function, r=2 Responses

In an example from Ries and Smith (1963), the choice of detergent brand (Brand M or X) is related to three other categorical variables: the softness of the laundry water (Softness soft, medium, or hard), the temperature of the water (Temperature high or low), and whether the subject was a previous user of Brand M (Previous yes or no). The linear response function, which could also be specified as RESPONSE MARGINALS, yields one probability, Pr(brand preference=M), as the response function to be analyzed. Two models are fit in this example: the first model is a saturated one, containing all of the main effects and interactions, while the second is a reduced model containing only the main effects. The following statements produce Output 29.1.1 through Output 29.1.4:

The PROC CATMOD statement is not required due to the interactive capability of the CATMOD procedure. The NOPROFILE option suppresses the redisplay of the "Response Profiles" table. The CLPARM option produces 95% confidence limits for the parameter estimates. Output 29.1.5 through Output 29.1.7 are produced.

The design matrix in Output 29.1.5 displays the results of the differential-effects modeling used in PROC CATMOD.

Output 29.1.5
Main-Effects Design Matrix

Detergent Preference Study

Main-Effects Model

The CATMOD Procedure

Data Summary

Response

Brand

Response Levels

2

Weight Variable

Count

Populations

12

Data Set

DETERGENT

Total Frequency

1008

Frequency Missing

0

Observations

24

Response Functions and Design Matrix

Sample

ResponseFunction

Design Matrix

1

2

3

4

5

1

0.41667

1

1

0

1

1

2

0.38182

1

1

0

1

-1

3

0.64179

1

1

0

-1

1

4

0.58427

1

1

0

-1

-1

5

0.41071

1

0

1

1

1

6

0.43103

1

0

1

1

-1

7

0.67143

1

0

1

-1

1

8

0.53922

1

0

1

-1

-1

9

0.48214

1

-1

-1

1

1

10

0.45690

1

-1

-1

1

-1

11

0.60417

1

-1

-1

-1

1

12

0.46226

1

-1

-1

-1

-1

The analysis of variance table in Output 29.1.6 shows that previous use of Brand M, together with the temperature of the laundry water, is a significant factor in whether a subject prefers Brand M laundry detergent. The table also shows that the additive model fits since the goodness-of-fit statistic (the residual chi-square) is nonsignificant.

Output 29.1.6
ANOVA Table for the Main-Effects Model

Analysis of Variance

Source

DF

Chi-Square

Pr > ChiSq

Intercept

1

1004.93

<.0001

Softness

2

0.24

0.8859

Previous

1

20.96

<.0001

Temperature

1

3.95

0.0468

Residual

7

8.26

0.3100

The chi-square test in Output 29.1.7 shows that the Softness parameters are not significantly different from zero; as expected, the Wald confidence limits for these two estimates contain zero. So softness of the water is not a factor in choosing Brand M.

Output 29.1.7
WLS Estimates for the Main-Effects Model

Analysis of Weighted Least Squares Estimates

Parameter

Estimate

StandardError

Chi-Square

Pr > ChiSq

95% Confidence Limits

Intercept

0.5080

0.0160

1004.93

<.0001

0.4766

0.5394

Softness

hard

-0.00256

0.0218

0.01

0.9066

-0.0454

0.0402

med

0.0104

0.0218

0.23

0.6342

-0.0323

0.0530

Previous

no

-0.0711

0.0155

20.96

<.0001

-0.1015

-0.0407

Temperature

high

0.0319

0.0161

3.95

0.0468

0.000446

0.0634

The negative coefficient for Previous () indicates that the first level of Previous (which is shown to be 'no') is associated with a smaller probability of preferring Brand M than the second level of Previous (with coefficient constrained to be 0.0711 since the parameter estimates for a given effect must sum to zero). In other words, previous users of Brand M are much more likely to prefer it than those who have never used it before.

Similarly, the positive coefficient for Temperature indicates that the first level of Temperature (which, from the "Population Profiles" table, is 'high') has a larger probability of preferring Brand M than the second level of Temperature. In other words, those who do their laundry in hot water are more likely to prefer Brand M than those who do their laundry in cold water.